Given the equation $$ \sqrt{3x + 12} = x + 6 $$, isolate the radical before solving for x.

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To isolate the radical in the equation $$ \sqrt{3x + 12} = x + 6 $$, we can follow these steps:

1. **Identify the radical**: The radical in the equation is $$ \sqrt{3x + 12} $$.

2. **Isolate the radical**: The radical is already isolated on the left side of the equation. Therefore, we can proceed to the next step.

Now, we can square both sides of the equation to eliminate the square root:

$$
(\sqrt{3x + 12})^2 = (x + 6)^2
$$

This simplifies to:

$$
3x + 12 = (x + 6)(x + 6)
$$

Next, we will expand the right side:

$$
3x + 12 = x^2 + 12x + 36
$$

Now, we can rearrange the equation to set it to zero:

$$
0 = x^2 + 12x + 36 - 3x - 12
$$

This simplifies to:

$$
0 = x^2 + 9x + 24
$$

Now, we will solve the quadratic equation $$ x^2 + 9x + 24 = 0 $$.

Let's proceed to solve this equation.
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using the quadratic formula in the complex numbers system:
  $$x^{2}+9x+24=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-9\pm \sqrt{9^{2}-4\times 24}}{2}$$
- step2: Simplify the expression:
  $$x=\frac{-9\pm \sqrt{-15}}{2}$$
- step3: Simplify the expression:
  $$x=\frac{-9\pm \sqrt{15}\times i}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{-9+\sqrt{15}\times i}{2}\&x=\frac{-9-\sqrt{15}\times i}{2}\end{align}$$
- step5: Simplify the expression:
  $$\begin{align}&x=-\frac{9}{2}+\frac{\sqrt{15}}{2}i\&x=\frac{-9-\sqrt{15}\times i}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=-\frac{9}{2}+\frac{\sqrt{15}}{2}i\&x=-\frac{9}{2}-\frac{\sqrt{15}}{2}i\end{align}$$
- step7: Rewrite:
  $$x_{1}=-\frac{9}{2}-\frac{\sqrt{15}}{2}i,x_{2}=-\frac{9}{2}+\frac{\sqrt{15}}{2}i$$
- step8: Remove the complex number(s):
  $$\textrm{No real solution}$$
The quadratic equation $$ x^2 + 9x + 24 = 0 $$ has no real solutions.

This means that the original equation $$ \sqrt{3x + 12} = x + 6 $$ does not have any real values of $$ x $$ that satisfy it.

To summarize:
- We isolated the radical and squared both sides.
- We derived the quadratic equation $$ x^2 + 9x + 24 = 0 $$.
- The quadratic has no real solutions, indicating that there are no real values of $$ x $$ that satisfy the original equation.
There are no real solutions for $$ x $$ in the equation $$ \sqrt{3x + 12} = x + 6 $$.

Given the equation \( \sqrt{3x + 12} = x + 6 \), isolate the radical before solving for x.

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